Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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1answer
22 views

Calculate the value of K in the matrix

If the matrix $A = \begin{pmatrix} 0 & 2 \\ K & -1 \end{pmatrix}$ satisfies $A(A^3+3I)=2I$ then value of K is: Answer: 1 Attempt: I thought of calculating polonium, and showing that the matrix ...
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0answers
15 views

Is it possible for a nonnormal magic square to have the same magic constant as a normal magic square of the same order?

I've been programming some quick and fast solutions to 4x4 magic square problems, and I've come across an assumption I've been using: The only squares with the magic constant 34 are the normal 4x4 ...
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0answers
12 views

Can we solve the matrices equations of vector based on a part of the vector itself after decomposition

Let's have the equation as following: $T \times x = s$ where $T$ is a unitary $N \times N$ matrix, and $x, s$ are vectors of dimensions $N \times 1$. Let's have $x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 ...
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0answers
18 views

Find basis of image and basis of Kernel of linear operator

Sorry if similar questions have been asked before but I don't really understand them. Question: Let $T$ be a linear operator in $\mathbb{R}^4\rightarrow\mathbb{R}^4$ such that $T\begin{pmatrix}1\\2\\1\...
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2answers
31 views

Why is $(GL(2,\mathbb{Z}_3),\cdot)$ a group?

I am doing an exercise where I have to use that $(GL(2,\mathbb{Z}_3),\cdot)$ is group. I just have a hard time accepting that this a group. $GL(2,\mathbb{Z}_3)$ is described as "the set of ...
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0answers
14 views

Matrix equation with simple solution in scalar case

I need to solve the following equation for $P \in\mathbb{R}^{r\times d}$ $$(PAP^\top)^{-1}P = G,$$ where the other quantities are known: $A\in\mathbb{R}^{d\times d}$, $G\in\mathbb{R}^{r\times d}$. If ...
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0answers
9 views

Given an lp-norm constrain on a matrix W, I wonder the largest value of the lq-norm of W where q is the dual of p.

let $W \in \mathbb{R}^{n \times m}$ and $\Vert \cdot \Vert_p$ be the $\ell_p$-norm of a matrix, and we have $p \ge 1, \frac{1}{p} + \frac{1}{p^*} = 1$, then what is the value (or the upper bound ...
2
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1answer
26 views

How to find the rotation matrix (with no x rotation) between two rotation matrices?

I need to find the rotation matrix (with no $x$ rotation) between two rotation matrices. Given a starting rotation matrix $\textbf{R}_a$ and a setpoint $\textbf{R}_{SP}$. I need to find the rotation ...
2
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1answer
68 views

Solution to a matrix quadratic

Let $\boldsymbol{X}$ be an $n \times m$ matrix and $\boldsymbol{A}$ be an $n \times n$ invertible matrix with $n > m$. I'm trying to find a solution to $$\boldsymbol{X}^\top \boldsymbol{A} \...
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0answers
12 views

What conditions need to be met for matrices $A$ and $B$ to guarantee the correlation?

Define $X\in R^{m\times m}$, $Y\in R^{n\times n}$ , $A\in R^{a\times m}$, $B\in R^{b\times n}$. If the correlation matrix (e.g. canonical correlation analysis) between $X$ and $Y$ is $\rho_ {X,Y}$, ...
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2answers
72 views

If product of two matrices is a diagonal matrix and one of them is diagonal matrix , then will other necessarily be a diagonal matrix?

I was asked to find the inverse of matrix $A=\text{diag}(a_1,a_2,...,a_n)$ Let $B$ be the inverse of $A$, then we have to find $B$ such that $AB=I$ where $I$ is identity matrix of same order as of $A$....
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0answers
25 views

Computing cyclic decomposition of $\mathbb{R}^3$ w.r.t. a matrix

I am trying to compute the cyclic decomposition of a $3\times 3$ nontriangular matrix $A$, whose minimal polynomial is not equal to the characteristic polynomial, which guarantees that the space $\...
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0answers
18 views

the range of eigenvalues of two real symmetric matrix's sum

$P, Q$ are two matrices (real and symmetric). The eigenvalues of $P$ are in $[p, q]$ and the eigenvalues of $Q$ are in $[r, s]$. What is the range of eigenvalues of $P+Q$? Intuitively it seems to be $[...
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1answer
57 views

I'm trying to find where a "3D logarithmic spiral" converges.

Background and problem It's in quotes because I don't know what I'm talking about. I'm more of an artist who likes to dabble in math so bare with me. I know the title is a little confusing so I have ...
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0answers
11 views

Notation to find the indices of the columns that contain the minimums of each row in a matrix

I am doing something that is easy in Python, but I need to document what I'm doing in a paper in a professional way. I have an $n x m$ matrix, $D$. In code, I pull out the column indices that contain ...
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3answers
68 views

How can I find A matrix who has A^3=?

My question is to find a matrix $A∈\mathbb{R}^{2\times2}$ for which $${A}^{3} =\begin{bmatrix} 64 & 64 \\ -9 & 16\end{bmatrix}^{-1}.$$ I tried to write this matrix like A^3=A^2*A after taking ...
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0answers
19 views

Determinant of two matrices that differ only in one column

There are two $n \times n$ matrices as follow (close to 5 diagonal): $$ A = \begin{bmatrix} a_{1}&b_{1}&c_{1}&0&0&0&0&0&0&\ldots&0\\ b_{1}&a_{2}&b_{2}&...
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2answers
118 views

Is it true that $\|A\| \leq \|A^2\|$ for $A\in SL(2, \mathbb{R})$?

Is it true that $$\|A\| \leq \|A^2\|$$ for $A \in SL(2,\mathbb{R})$, where $\| \|$ is the operator norm that is the first singular value? $$\left \| A \right \| =\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\...
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1answer
16 views

How does the definition of the regulator work?

If we create a $r x (r+1)$ matrix with entries $N_jln(|\sigma_j(u_i)|)$ with $N_j$ being 1 if $\sigma_j$ is a real embedding and 2 if its complex, any $r x r$ submatrix generated by deleting a column ...
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0answers
18 views

determine the transformation matrix $A$ with respect to the ordered basis $B = \{(1,1,1),(1,2,1),(1,0,0)\}$

i refer to the exercise in chapter 2 2.19 b https://mml-book.github.io/book/mml-book.pdf i do not have any clue about this kind of task, i solved a by myself but here it stops enter image description ...
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0answers
33 views

How to derive an inverse of Gaussian Kernel

As an example, say I have a 2D function (Gaussian process kernel): $$K(x_i, x_j) \equiv \exp(-\alpha |x_i-x_j|^2)+\beta \delta_{ij}$$ Is there a way to analytically express $K^{-1}(x_i,x_j)$, s.t. ...
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2answers
27 views

How do you Derive $\Delta x = -\textbf{H}^{-1}\textbf{g}$ from Quasi-Newtonian Methods?

The equation: $$\Delta x = -\mathbf{H}^{-1}\mathbf{g}$$ Is the key part of Quasi-Newtonian methods, where $\Delta x$ is the coordinate shift required to reach a minimum, $\textbf{H}$ is the hessian ...
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2answers
63 views

Showing that the product of two permutation matrices results in another permutation matrix [closed]

The idea that the product of two permutation matrices gives another permutation matrix makes sense to me, since we know that they only have one entry of 1 in each row and column (and 0s everywhere ...
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0answers
44 views

If ${\bf A} - {\bf B}$ is positive semidefinite, then is ${\bf A}^{1/2} - {\bf B}^{1/2}$ still positive semidefinite? [duplicate]

For two real symmetric and positive semidefinite matrices, ${\bf A}$ and ${\bf B}$, of same size, can we prove ${\bf A} \succeq {\bf B} \quad \iff \quad {\bf A}^{1/2} \succeq {\bf B}^{1/2}$, where $\...
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0answers
23 views

Find change of coordinates matrix given two basis [closed]

How do I find the change of coordinates matrix from B` to B? Thank you! $$ \beta = \{x^2 - x +1; ~x+1,~x^2 +1\} $$ and $$ \beta' = \{ x^2 + x + 4,~4x^2 - 3x +2,~2x^2+3 \} $$
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1answer
25 views

Converting recursive equation into matrices

Here is example of converting fibonacci function into matrices. Fibonacci sequence defines $$ f(1)=1 $$ $$ f(2)=1 $$ $$ f(x) = f(x-1) + f(x-2) $$ It can be converted into matrix $$ \begin{bmatrix} 1 &...
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1answer
32 views

What is the Hessian matrix of $L(\beta)=\sum_{i=1}^n[y_ix_i^T\beta-\exp(x_i^T\beta)]$?

Given a covariate vector $x_i=[x_{i1}, x_{i2}, \dots, x_{ip}]^T \in R^{p\times 1}$ and associated labels $y_i\in R$. We have a parameter $\beta=[\beta_1, \beta_2, \dots, \beta_p]^T$. For $$L(\beta)=\...
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0answers
32 views

How to get this result $\frac{\partial a^Tx}{\partial x^T}=a^T$?

For a n by 1 column vector $x$ and $a$, we know that $$\frac{\partial a^Tx}{\partial x}=\frac{\partial x^Ta}{\partial x}=a$$ But I guess we have that $$\frac{\partial a^Tx}{\partial x^T}=a^T$$ How to ...
1
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0answers
47 views

Prove the existence of a similar matrix

$A$, $B$ are two matrices such that $A\ge0$ and $B\ge0$ and either $A>0$ or $B>0$. I am trying to show that matrix $BA$ is similar to a matrix with non-negative diagonal elements. Here; $A$ and $...
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0answers
23 views

How to find normal vector direction on a time-evolving curve with curvature and torsion?

I have a question about computing normal vector direction to a time-evolving curve please. The normal vector direction for a curve in 3D space with curvature k and zero torsion can be found by ...
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0answers
12 views

Given that A = ( 2 2 1 3 ) and B = ( 1 1 1 2 ), find (a) the matrix P such that AB = P. (b) P -1 [closed]

All the working without missing a detail such as showing the inverse
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2answers
66 views

Can $n\times n$ matrix have algebraic multiplicity less than $n$?

I can't seem to think of an example that would have an algebraic multiplicity less than $n$.
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0answers
17 views

Diagonalizable matrix inspection from its row echelon form

I have a question regarding when a matrix is diagonalizable Suppose we have a $n \times n$ matrix $A$. Let's say that I row reduce and I find the matrix $B$, which is the row echelon form of the ...
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0answers
15 views

I wanted to know if this LU Factorization was correct. I don't have anyone else to ask for feedback.

Compute the LU factorization of the following matrix. No row interchanges will be necessary. You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix....
3
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1answer
57 views

Light bulbs in a rectangular grid

Suppose we have a grid of $5 \times 5$ light bulbs with a switch to each bulb. If we press a switch it toggles all the lights in its row and column. Given any initial configuration, is it possible to ...
0
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1answer
17 views

Calculating matrices in the following question

If $A^2 = I, B^2 = \begin{bmatrix}3 & 2\\-2 & -1 \end{bmatrix}, AB = \begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$, find $BA$. What I did was the following: $A^2 = I ; AA^{-1} = I {\...
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1answer
38 views

Proving a matrix in invertible from the following question

The question says: If matrix $A$ is invertible and $A + B = AB$, prove that matrix $B$ is invertible and $A^{-1} + B^{-1} = I$ Firstly, I was thinking that we can only prove that matrix $B$ is a ...
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0answers
42 views

Proving the equality in the attached Image

I am trying to learn the mathematics behind Finite elements as an engineer. can anyone help me with the proof below ? I have almost figured out the solution. The problem is I get a multiplication of ...
2
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1answer
43 views

Show that the matrix is a generator matrix of a MDS code

Let $a_1,\dots,a_n$ be pairwise distinct elements of $\mathbb{F}_q$ and $k ≤ n$. I have to show that the matrix $\begin{bmatrix}1&\dots&1&0\\a_1&\dots&a_n&0\\a_1^2&\dots&...
1
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1answer
29 views

Matrix/vector multiplication

I have the following vector $z=\begin{pmatrix} x \\ y \end{pmatrix}$. I also have the function \begin{equation} f=\begin{pmatrix} -5\beta xy \\ 5\beta xy \end{pmatrix} \end{equation} I need to rewrite ...
0
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1answer
63 views

Find $5\times5$ invertible matrix $A$ over $\mathbb{F}_3$ such that $A^{-1} = 2A^3 +2I$, $A \neq I$

I have tried to solve the above using the cayley hamilton theorem which yields nothing as I get $-1$ which is not in my field. I feel like I need to do a sub-block decomposition.
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0answers
16 views

Combining two linear matrix equations with non-square linear operators?

Assume I have two equations: $$\mathbf{x} = \mathbf{A}\mathbf{y}$$ where $\mathbf{x} \in \mathbb{R}^{m}$, $\mathbf{y} \in \mathbb{R}^{n}$, and $\mathbf{A}$ is a matrix of size $m$-by-$n$, as well as $$...
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1answer
22 views

Bounding the operator norm of a matrix after multiplication and taking inverses.

Suppose $D$ is a $n$ by $n$ diagonal matrix with positive entries on the diagonal, and $A$ is a $m$ by $n$ matrix, is there anything we can say about the operator norm (or value of maximum entry) of ...
1
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1answer
49 views

How to show that $\varphi$ is a Group Homomorphism between two groups

Consider the map $\varphi:\text{GL}(2,\mathbb{Z}_3)\rightarrow \mathbb{Z}_3^*$, where $\varphi(M)=\text{det}(M)$ I have to show that $\varphi$ is a group homomorphism between the groups $(\text{GL}(2,...
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0answers
20 views

Characterization of representation matrix of alternating bilinear form

Sorry for my bad English. Let $K$ be a field s,t, ${\rm ch} K=2$, and $V$ be $K$-vector space of dimension $n$, and take the basis $v_1, \dots, v_n\in V$. Then bilinear form $f: V\times V\to K$ is ...
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0answers
43 views

How can I find a matrix $A∈\mathbb{R}^{2\times2} $ for which ${A}^{3} =\begin{bmatrix} 64 & 64 \\ -9 & 16\end{bmatrix}^{-1}$? [closed]

My question is to find a matrix $A∈\mathbb{R}^{2\times2}$ for which $${A}^{3} =\begin{bmatrix} 64 & 64 \\ -9 & 16\end{bmatrix}^{-1}.$$ Thanks.
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0answers
21 views

Are permutation cycles equivalent to each other with respect to other permutations?

Let $M$ be a $0/1$ matrix where $(i,i+1)$ entries are $1$ if $i\in\{1,\dots,n-1\}$ and $(n,1)$ entry is $1$ and remaining entries are $0$. If $P$ and $Q$ are other permutation matrices when is $PMQ$ ...
0
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1answer
41 views

Finding matrix $X$ with given $\cos(X)$ value [closed]

My question is: Find matrix $X$ with given $\cos(X)$ value. I tried to solve with this equation $aX+bI=\cos(X)$. with $\cos^{-1}(A)=X$ but doesn't work. Find a matrix $X \in \mathbb R _ { 2 \times ...
1
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1answer
49 views

Matrix identity over a general field

This is a problem from Hoffman and Kunze: Let $A$, $B$, $C$, $D$ be $n \times n$ matrices over an arbitrary field $F$ that commute with each other, then the determinant of the the $2n \times 2n$ ...
2
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2answers
76 views

Multiplicity of factors of a determinant by observation

Consider the determinant $$ \Delta = \begin{vmatrix} x+1 & 3 & 5 \\ 2 & x+2 & 5 \\ 2 & 3 & x+4 \end{vmatrix}$$ It can be explicitly shown to be $$\Delta = (x-1)^2(x+9)$$ ...

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